I would actually like to echo @JiK a bit, especially since I am a physicist.
One of my first lessons in physics was on the fundamental concept of measure and measurement. We we tasked with defining and coming up with our own concept of length. We split into groups and those of us who paid attention to the lecture found a convenient object to define as a reference unit. Some chose pens or markers. Others chose erasers. Some didn't choose anything. One student who must've had a parent who studied physics chose a flashlight and a stopwatch.
Once chosen we were each tasked with measuring a length on the white-board without disturbing or accidentally smudging it and the figure out how to convert from group-A's pen-units to group-B's eraser-units. The group with the flash-light could not proceed, lacking the precision required to measure nano-seconds on their stop-watch.
To find the conversion rates, the goal is to then go to each group and have them make a 'number-line' where each tick-mark on the line represents one unit of eraser/pen/marker/coin/etc and carefully aligning the lines and marks to determine that 10 erasers is 3 pens or 20 coins (as an example, exact numbers must be obtained yourself). To get really accurate conversion ratios, the lines must be drawn out not just on a single paper, but across many. This allows the students to see that in fact 100 eraser-tick-marks can fit 31 pen marks and 199 coin marks. To get 'perfect' measurements of conversion ratios the number of tick marks that must be made and accurately counted becomes endless. In other words, despite the best tools and technologies available to high school students there is no such thing as a perfect ratio.
The teacher then stole one eraser-unit and broke it. Then he asked the class if the device could still produce accurate measurements. Some said no. Some said yes, but you'd have to glue it back together. The teacher then went up to the board and measured the drawn line with the broken pen and counted out the number of ticks that the broken piece could make just as had been done with the eraser before it was split. Of course the tick marks were different, but the broken-eraser-unit still produced a measurement which could be put into ratio with the unbroken-eraser-unit and the pen-unit.
At the end the teacher measured the length of the line in centimeters and converted that to pen-units.
I suppose my take-away was that numbers in the process of measurement are tools, or perhaps a better term is instructions, that tell the reader how to get a specific quantity. A number on it's own doesn't mean anything. There is no length represented by '5' any more so than there is a mass represented by '10'. The same number can represent different lengths because the other half of a physically real measurement is the unit.
Physically real quantities are more than just numbers. They are numbers and a physical-object. A directed-displacement is numbers and vectors. Energy is a number and a unit of work. Delays are a number and a length of time.
Are numbers themselves real? Sure, insofar as a set of instructions are or words in general. But they do not exist independently of people or their perception and understanding. Integers certainly exist in the conception of even the like of bees, ants, birds, dogs, humans, and fish. But fractions (ratios) require another level of abstraction. Not just the ability to count, but the ability to communicate the count and determine what that count means in a different context (covert from Ant-A steps to Ant-B steps). Real numbers require yet more abstraction.
I think the most meaningful answer is a rhetorical question: "Do you know what [ concept ] is?" If you can say yes and accurately use the concept, then yes you think it exists. If you cannot say yes or cannot accurately use it then you do not think it exists. Perhaps the better expression is to say that "The act of creating an idea makes it real for you." Surely the physical distance between object exists independently of humans or anything else, but the number of steps to cross that distance is dependent on people.
Others mention unit-less numbers, but all numbers in physics come from ratios of units. That is they are made from physical objects and physically real distances and masses and delays and energies and ... I don't think the fine structure constant is any more real than any other number, especially since it is a ratio of real quantities, but even more to the point it isn't even exactly 1/137. It is approximately 1/137.
Others point to π or e as being real, but these are still ratios of physical objects. Additionally, drawing a circle on a curved surface increases or decreases the value you get for π depending on the geometry of the surface (positive, negative, or zero curvature). On a sphere the sphere-π (which is obtained from the ratio of the circumference to the radius, not of the sphere but the circle on the sphere) is less than plane-π. On a negatively curved surface, the saddle-π is greater than plane-π. These are all still ratios of lengths.
I will end this thought here.